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<h3 class="heading"><span class="type">Paragraph</span></h3>
<p>Here we use (<code class="code-inline tex2jax_ignore">[cross-reference to target(s) "def" missing or not unique]</code>) directly: eg</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
{\rm For} \ f(t)=1, \ \ {\mathcal L}[1] = \int_0^\infty e^{-st}  \, dt \ =
\left[-\frac{1}{s}e^{-st}\right]_0^\infty= \frac{1}{s}.
\end{equation*}
</div>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
{\rm For} \ f(t)=t, \ \ {\mathcal L}[t] = \int_0^\infty e^{-st} t \, dt \ =
\left[-\frac{1}{s}e^{-st}t\right]_0^\infty + \int_0^\infty \frac{1}{s}e^{-st}=
\frac{1}{s^2}.
\end{equation*}
</div>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
{\rm For} \ f(t)=y^{\prime}(t), \ \ {\mathcal L}\left[f(t)\right] =
\int_0^\infty e^{-st} \frac{dy}{dt} \, dt \ = \left[e^{-st} y \right]_0^\infty +
\int_0^\infty s e^{-st} y \, dt \ = -y(0)+s F(s),
\end{equation*}
</div>
<p class="continuation">where <span class="process-math">\(F(s)={\mathcal L}(y(t))\text{.}\)</span> More generally,</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
{\mathcal L}(y^{(n)})=s^n F(s)-s^{n-1} y(0)- \cdots -s y^{(n-1)}(0)-y^{(n-1)}(0).
\end{equation*}
</div>
<p class="continuation">For <span class="process-math">\(f(t)=e^{at}, \ a \ {\rm constant},\)</span></p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
{\mathcal L}[e^{at}] = \int_0^\infty e^{-st} e^{at} \, dt \ =
\int_0^\infty e^{-(s-a)t}  \, dt \ =
\left[-\frac{1}{s-a}e^{-(s-a)t}\right]_0^\infty= \frac{1}{s-a}, \ \ s&gt;a.
\end{equation*}
</div>
<span class="incontext"><a href="sec8_2.html#p-437" class="internal">in-context</a></span>
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